Margin of Error and Hypothesis Tests

Sample proportion and sampling variability

A population is the whole group we want to know about. A sample is the part of it that we actually ask. A random sample gives every member of the population the same chance of being picked.

The sample proportion is the fraction of the sample with the feature we are interested in:

We usually write it as a percentage. Different random samples give different proportions. This is called sampling variability. It is not a mistake. It is just chance at work.

The margin of error

How big is the typical gap between a sample proportion and the true population proportion? It depends on the sample size $n$. Bigger samples vary less.

Notice the effect of $n$. To halve the margin of error you need four times the sample size.

Choosing the sample size

Sometimes we work backwards. A council wants a margin of error of no more than $3\%$. How many people must it ask?

Hypothesis tests for a proportion

A hypothesis test uses a sample to decide whether a claim about a population is believable.

• The null hypothesis, $H_0$, is the claim being tested. For example, "the coin is fair", so the proportion of heads is $50\%$.

• The alternative hypothesis, $H_1$, is what we accept if we reject $H_0$. For example, "the coin is not fair".

Always state the conclusion in context. Say "there is no evidence that the coin is unfair", not just "accept".

Comparing polls and judging reliability

Interpreting an interval is as important as finding it.

Majority claims. A claim that "the majority" support something means more than $50\%$. It is supported only if the whole interval lies above $50\%$. For the bicycle survey the interval was $50.7\%$ to $57.3\%$, which is entirely above $50\%$, so a majority is supported (just).

Consistent polls. Two polls are consistent if their intervals overlap. Then both could be estimating the same true value.

Finding the sample size that rejects a claim. A candidate claims $50\%$ support, but a sample shows $56\%$. The claim is rejected when $50\%$ falls outside $56\% \pm \dfrac{1}{\sqrt{n}}$. That happens when $\dfrac{1}{\sqrt{n}} < 6\% = 0.06$, so $\sqrt{n} > 16.67$ and $n > 277.8$. The smallest whole number is $n = 278$. With a smaller sample the interval is wide enough to include $50\%$, so the claim cannot be rejected.

A Type I error means rejecting a null hypothesis that is actually true. In the candidate example, it would mean deciding she does not have $50\%$ support when she really does. At the $5\%$ level, this happens about $5\%$ of the time.

Back to the start: $54$ out of $100$ gives a margin of error of $10\%$, so the interval is $44\%$ to $64\%$. That includes values below $50\%$, so the radio station cannot claim a majority.

Summary